How to make this object child of another object, and conform to a curve at once?
You can't do it this way, because an object with a Curve modifier will only correctly follow the curve if you move it on one of the 3 straight axis (X, Y or Z). If you move it on another axis, which happens if you ask him to follow an object which follows a curve, it will go messy.

I see at least 2 solutions, but maybe there are simpler ones?

1) Use a Curve modifier for your tail

In that case you must deparent the tail from the asteroïd. The asteroïd will follow the curve with a Follow Path constraint, and the tail will follow the curve with a Curve modifier. You must make sure that the tail have the same speed as the asteroïd, as if it was parented to it even if it's not, so you have to play with the Dopesheet keyframes and the Graph editor parameters.



2) Create an armature for your tail, that will follow the path


Create an armature with several bones and with the same length as your tail.
Parent your tail to this armature and place the armature at the beginning point of your curve.
Create an Empty a this beginning point, give it a Follow Path constraint with the curve as the Target.
Parent your armature to this Empty.
Duplicate this Empty with shift D + Enter, then, in its Follow Path constraint, change the Offset so that from Empty 0 to Empty 1 there will be the length of one of your armature bones.
Repeat the Duplicate + Offset operations so that you have as many Empty as your number of bones (+ the parent Empty)
Select the armature and, in Edit mode, give each bone a Track To constraint, with the right Empty as the Target.
Move each Empty along the curve with the Follow Path Offset, so that they are exactly at the position of their bone's head.
You can switch your armature from Octahedral display to B-Bones, and divide each bone in several Segments to smooth the shape of your tail.